A324525 Numbers divisible by prime(k)^k for each prime index k.

1, 2, 4, 8, 9, 16, 18, 27, 32, 36, 54, 64, 72, 81, 108, 125, 128, 144, 162, 216, 243, 250, 256, 288, 324, 432, 486, 500, 512, 576, 625, 648, 729, 864, 972, 1000, 1024, 1125, 1152, 1250, 1296, 1458, 1728, 1944, 2000, 2048, 2187, 2250, 2304, 2401, 2500, 2592

Offset

1,2

Comments

These are a kind of self-describing numbers (cf. A001462, A304679).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.

Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - Amiram Eldar, Nov 23 2020

Example

The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   54: {1,2,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}

Maple

q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
select(q, [$1..3000])[];  # Alois P. Heinz, Mar 08 2019

Mathematica

Select[Range[1000], And@@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>k>=PrimePi[p]]&]
seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* Amiram Eldar, Nov 23 2020 *)

Crossrefs

Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.

Cf. A109298, A324524, A324571, A324572, A324587, A324588.

Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Sequence in context: A335851 A028982 A320137 * A175338 A071601 A114400

Adjacent sequences: A324522 A324523 A324524 * A324526 A324527 A324528

Keyword

nonn

Author

Gus Wiseman, Mar 08 2019

Status

approved